Time-Dependent Accretion Disks with Magnetically Driven Winds: Green's Function Solutions

TL;DR

This work derives exact Green's function solutions for the time-dependent evolution of a geometrically thin, Keplerian accretion disk with magnetically driven winds, allowing ν to scale as $ν \propto r^{n}$ and incorporating wind-related angular momentum and mass loss via parameters $\\psi$ and $\\lambda$. Green's functions are constructed for three inner-boundary conditions (zero torque, zero mass accretion rate, and finite torque with finite accretion) and for both $r_{\rm in}=0$ and $r_{\rm in}>0$, using Hankel and Weber-type transforms to relate mode weights to the initial surface density, including a Dirac-delta initial profile. A key result is the late-time evolution $\\dot{M},\\dot{M}_{\rm w} \propto t^{-(1+l)}$, where $l=\\sqrt{(1+\\psi)^2+4\\psi/(\\lambda-1)}}/(4-2n)$, with winds steepening decays and reducing disk lifetimes; boundary-condition effects are most pronounced for weak winds ($\\psi\lesssim1$) but become negligible when winds are strong ($\\psi \gtrsim 10$). The framework is applied to protoplanetary disks, revealing wind-driven evolution tracks in the $\\dot{M}$–$M_{d}$ and $(\\dot{M}+\\dot{M}_{w})$–$M_{d}$ planes and linking disk lifetime to wind strength, providing a robust tool to interpret accretion variability and disk dispersal timescales in wind-dominated regimes.

Abstract

We present Green's function solutions for a geometrically thin, one-dimensional Keplerian accretion disk that includes angular momentum extraction and mass loss due to magnetohydrodynamic (MHD) winds. The disk viscosity is assumed to vary radially as $ν\propto r^{n}$. We derive solutions for three types of boundary conditions applied at the inner radius $r_{\rm in}$: (i) zero torque, (ii) zero mass accretion rate, and (iii) finite torque and finite accretion rate, and investigate the time evolution of a disk with an initial surface density represented by a Dirac-delta function. The mass accretion rate at the inner radius decays with time as $t^{-3/2}$ for $n = 1$ at late times in the absence of winds under the zero-torque condition, consistent with Lynden-Bell \& Pringle (1974), while the presence of winds leads to a steeper decay. All boundary conditions yield identical asymptotic time evolution for the accretion and wind mass-loss rates, though their radial profiles differ near $r_{\rm in}$. Applying our solutions to protoplanetary disks, we find that the disk follows distinct evolutionary tracks in the accretion rate-disk mass plane depending on $ψ$, a dimensionless parameter that regulates the strength of the vertical stress driving the wind, with the disk lifetime decreasing as $ψ$ increases due to enhanced wind-driven mass loss. The inner boundary condition influences the evolution for $ψ< 1$ but becomes negligible at higher $ψ$, indicating that strong magnetically driven winds dominate and limit mass inflow near the boundary. Our Green's function solutions offer a general framework to study the long-term evolution of accretion disks with magnetically driven winds.

Figures (12)

• Figure 1: Radial profiles of the surface density $\Sigma$ (panel a), mass accretion rate (panel b), and integrated wind mass-loss rate (panel c) for a disk with a zero-torque boundary condition at the inner edge ($r_{\rm in} = 0$). Results are shown for two values of the wind-driving parameter: $\psi = 0$ (solid lines; no wind) and $\psi = 10$ (dashed lines; with wind). The blue, red, green, and purple lines correspond to $t / t_{\nu}(r_0) = 10^{-4}$, $0.01$, $0.1$, and $1$, respectively. The magnetic lever arm parameter is fixed at $\lambda = 3/2$. The quantities $\Sigma_0$, $r_0$, and $\dot{M}_0 = 3 \pi \nu(r_0)\Sigma_0$ are arbitrary normalization constants. In panel (a), the surface density profiles for $\psi = 0$ (blue solid) and $\psi = 10$ (blue dashed) coincide at $t = 10^{-4} t_{\nu}(r_0)$. Note that in panel (c), solid lines are absent because $\psi = 0$ corresponds to the no-wind case.
• Figure 2: The time evolution of the mass accretion rate, measured at $r = 10^{-5} r_0$ (inner region), and its corresponding slope, $n_{\rm acc} = {\text{d}} \ln \dot{M} / {\text{d}} \ln t$, are shown in panels (a) and (b), respectively, for a disk with a zero-torque boundary condition at the inner edge ($r_{\rm in} = 0$). Panels (c) and (d) present the total wind mass loss rate, integrated up to $r = 100 r_0$, and its slope, $n_{\rm w} = {\text{d}} \ln \dot{M}_{\rm w} / {\text{d}} \ln t$, respectively. Different colors represent different values of $\psi$, with $\psi = 0$ corresponding to a disk without wind. The dashed lines in panels (b) and (d) indicate a slope of $-(1 + l)$, as predicted by equation (\ref{['eq:mdotztor']}) for the late-time behavior of the mass accretion rate.
• Figure 3: The same format as in Figure \ref{['fig:ztrin0']}, but for a disk with a zero-torque boundary condition at a finite inner edge. The inner radius of the disk is set to $r_{\rm in} = 0.1r_0$.
• Figure 4: Panels (a), (b), (c), and (d) follow the same format as in Figure \ref{['fig:ztrin0_1']}, but correspond to a disk with a zero-torque boundary condition at a finite inner edge. The inner radius is set to $r_{\rm in} = 0.1 r_0$. Panels (e) and (f) show the bolometric luminosity, integrated from the inner edge to $r = 100r_0$, and its slope, $n_{\rm l} = {\text{d}} \ln L / {\text{d}} \ln t$, respectively. The dashed lines in panels (b), (d), and (f) represent a slope of $-(1 + l)$. The normalization constants $\dot{M}_0$ and $L_0$ are given by equation (\ref{['eq:mdot0']}) and equation (\ref{['eq:lum0']}), respectively.
• Figure 5: The same format as in Figure \ref{['fig:ztrinf']}, but for a disk with a zero mass accretion rate at a finite inner edge. The inner radius of the disk is set to $r_{\rm in} = 0.1 r_0$.